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Remapping in $\omega_0$

As the variable $\omega$ is replaced with $\omega_0$ in the integral

\int d\omega p(\omega,k_y,k_h)=
\int d\omega_0 {\left[{{d \omega} \over {d \omega_0}}\right]} 

each value of $\omega_0$ in the integral needs to be associated with the appropriate value of the field $p(\omega,k_y,k_h)$ which is $p(\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}},k_y,k_h)$.The new field $p^*(\omega_0,k_y,k_h)$ represents a remapping of the original field $p(\omega,k_y,k_h)$.Each value in the new field $p^*(\omega_0,k_y,k_h)$ with coordinates $(\omega_0,k_y,k_h)$ corresponds to the value in the field $p(\omega,k_y,k_h)$ with coordinates $(\omega={\omega_0 \sqrt{1+{{v_h^2} \over {\omega_0^2-v_y^2}}}}

The $\omega \rightarrow \omega_0$ remapping can be understood easier if one takes the particular case ky=0. The change of variable in $\omega$ (5) becomes


the Jacobian

{\left[ {{d \omega} \over {d \omega_0}} \right]}=
{{\mid {{2...
 ...over v} \mid} \over 
{\sqrt{{{4\omega_0^2} \over v^2}+k_h^2}}},\end{displaymath}

and the zero-offset migrated field becomes  
p_0(\omega_0,k_y=0)=\int dk_h 
{{\mid {{2\omega_0} \over v} ...
 ...v \over 2}
{\sqrt{{{4\omega_0^2} \over v^2}+k_h^2}},k_y=0,k_h).\end{displaymath} (10)
Compare equation (10) with the Gazdag (1978) and Stolt (1978) zero-offset migration equations:  
p(z,k_y) & = & \displaystyle{
\int d\ome...
 ...n(\omega_0){v \over 2}\sqrt{\omega_0^2+k_y^2},k_y)}.\end{array}\end{displaymath} (11)
Notice that the Stolt migration formula is comprised of an inverse Fourier transform and a remapping (interpolation). Equation (10) inside the integral in kh has the same form as the remapping in the Stolt formula, save for a constant coefficient $v \over 2$. This similitude suggests the equivalent formulation:  
p_0(t_0,k_y=0) & = & \displaystyle{
p(\omega,k_y=0,k_h)}.\end{array}\end{displaymath} (12)
The validity of the new equation is verified by replacing in equation (12) the exponential expression


by a new variable $\omega_0$ to get back the inverse Fourier transform of the initial expression.

The integration in kh represents the inverse Fourier transform at zero offset (h=0). However because equation (12) actually performs summation along hyperbolas in the spatial coordinate h, replacing the integral in kh by the inverse Fourier transform will not return the constant-offset sections.

Applying the same technique to the case $k_y \neq 0$ I obtain  
\int dk_h \int d\omega \;
p(\omega,k_y,k_h)\end{displaymath} (13)
which can be verified by substituting the exponential expression in $(\omega,v_y,v_h)$ with a new variable $\omega_0$defined as
\omega_0^2 = {1 \over 2} 
\left [ {\omega^2+v_y^2-v_h^2 + 
{\sqrt {(\omega^2+v_y^2-v_h^2)^2-4 \omega^2 v_y^2}}} \right ].\end{displaymath} (14)

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